Q111 / 160 · TS EAMCET 2022 (19 Jul Shift 2) · Mathematics

Assertion : The curves $y^2=4 x$ and $x^2=-2 y$ intersect at $(1,2)$ orthogonally. Reason : If the product of the slopes of the tangents drawn to two curves at their point of intersection is -1 , then the curves are said to cut each other orthogonally. The correct option among the following is

  • A
    (A) is true, (R) is true and (R) is the correct explanation for (A)
  • B
    (A) is true, (R) is true but (R) is not the correct explanation for (A)
  • C
    (A) is true but (R) is false
  • Correct answer
    D
    (A) is false but (R) is true

Solution

Given curves are $\mathrm{y}^2=4 \mathrm{x}$ and $\mathrm{x}^2=-2 \mathrm{y}$ intersect at $(1,2)$ orthogonally, for this slopes of $c^1=y^2-x=0$ and $\mathrm{C}_2=\mathrm{x}^2+2 \mathrm{y}=0$ have slopes $\mathrm{M}_{\mathrm{c}_1}$ and $\mathrm{M}_{\mathrm{C}_2}$ then $M_{c_1} \times M_{c_2}=-1$ here $M_{c_1}=\frac{1}{2}$ and $M_{C_2}=-1$ $\Rightarrow \mathrm{M}_{\mathrm{c}_1} \times \mathrm{M}_{\mathrm{c}_2} \neq-1$ So, Asseration (A) is false. (B) Reason is correct. Hence correct option is (d)

Hence, the correct answer is option: 'D'.

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